====== Pairwise 1-State ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)

Here we consider an extension to the [[..:simple_1-state:start]] model where we relax the //simplicity// criteria.

===== Pairwise =====

//Pairwise// reactions satisfy the following criteria:
  - The assumptions of [[wp>reaction kinetics]] apply.
  - All reactions have a total [[wp>order of reaction]] less than three.  There can be zero (nil), one (solo), or two (pair) reactants involved in a reaction.  We assume higher order reactions are negligibly rare or can be decomposed into sets of lower order reactions.

Unlike the //simplicity// condition we **do not** require:
  - All reactions involve change to only one reactant.  The product set differs from the reactant set by exactly one individual.  We assume reactions with more than one reactant changing are composed of stages where one reactant changes at a time.
  - Every reactant/product represents a single individual in the population.  Bonded states are not allowed.

Instead, any product set is permitted.  Our only restriction is that reactions involve no more than two reactants.

===== States =====

States refers to the number of distinguishable states of the individuals in the population.  A state may be a species or a member's gender or physiological state.  In this 1-state model individuals are completely indistinguishable.  There exists only one species and no distinction between members of the species.

===== Reactions =====

In the case of a simple 1-state system this means there are three groups of reactions, characterized by the possible reactant sets:
\[
\begin{array}{rcll}
  \emptyset & \xrightarrow{\alpha_i} & a_i N & \text{(no reactants)} \\
  N   & \xrightarrow{\beta_i}        & b_i N & \text{(one reactant)} \\
  2 N & \xrightarrow{\gamma_i}       & c_i N & \text{(two reactants).}
\end{array}
\]
where \(i\) is an integer index and \(a_i, b_i, c_i\) are non-negative integers.

For example, we can recover the [[..:simple_1-state:start]] model with the following mapping:
\[
\begin{array}{rclcrl}
  \emptyset & \xrightarrow{\alpha} & N         & : & \alpha_1 = \alpha,   & a_1 = 1 \\
  N   & \xrightarrow{\beta}        & 2 N       & : & \beta_1 = \beta,     & b_1 = 2 \\
  2 N & \xrightarrow{\gamma}       & 3 N       & : & \gamma_1 = \gamma,   & c_1 = 3 \\
  N   & \xrightarrow{\delta}       & \emptyset & : & \beta_2 = \delta,    & b_2 = 0 \\
  2 N & \xrightarrow{\epsilon}     & N         & : & \gamma_2 = \epsilon, & c_2 = 1.
\end{array}
\]

===== Dynamics =====

The population density, \(n\), will behave as
\[
  \diff{n}{t} = A + b n + c n^2
\]
where the constants \(A, b,\) and \(c\) are composed of the reaction parameters as follows:
\[
\begin{array}{}
  A & \equiv & \sum_i \alpha_i a_i \geq 0 \\ 
  b & \equiv & \sum_i \beta_i (b_i - 1) \\
  c & \equiv & \sum_i \gamma_i (c_i - 2)
\end{array}
\]
(the constant \(A\) is capitalized as a reminder that it is strictly non-negative.

==== Flow ====

To describe the full dynamics draw the (quadratic) rate \(\diff{n}{t}\) versus \(n\). Where the rate is positive \(n\) is growing, negative indicates declining, and a zero rate marks an equilibrium.

The particular values of the constants \(A, b,\) and \(c\) will determine where these different regimes occur and hence the overall dynamics.

Note that the analytic results should be considered against biological plausibility.  For example, if the population is determined to diverge then the model is inadequate--clearly something else must be going on since real populations don't diverge.  Similarly, if the population is unstable it is less biologically relevant because most empirical populations we are able to track tend to persist for an extended period.